# On the distributions of scan statistics of a two-dimensional Poisson
process

#### by

## Sven Erick Alm

#### Uppsala University

##### In Advances in Applied Probability (SGSA) 29, 1-18 (1997).

### Abstract

Given a two-dimensional Poisson process, *X*, with intensity $\lambda$,
we are interested in the largest number of points, *L*, contained in a
translate of a fixed scanning set, *C*, restricted to lie inside a
rectangular area.

The distribution of *L* is accurately approximated for rectangular
scanning sets, using a technique that can be extended to higher dimensions.

Reasonable approximations for non-rectangular scanning sets are also
obtained using a simple correction of the rectangular result.

#### Key words:

Poisson process, scan statistic, approximation

Research supported by the Axel and Margaret Ax:son Johnson Foundation.
1998-04-06, Sven Erick Alm, sea@math.uu.se